## Discrete approximations to the double-obstacle problem and optimal stopping of tug-of-war games

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- by Luca Codenotti, Marta Lewicka and Juan Manfredi PDF
- Trans. Amer. Math. Soc.
**369**(2017), 7387-7403 Request permission

## Abstract:

We study the double-obstacle problem for the $p$-Laplace operator, $p\in [2, \infty )$. We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-of-war games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the step-size controlling the single shift in the token’s position, converges to $0$. We propose a numerical scheme based on this observation and show how it works for some examples of obstacles and boundary data.## References

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## Additional Information

**Luca Codenotti**- Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- Email: luc23@pitt.edu
**Marta Lewicka**- Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- MR Author ID: 619488
- Email: lewicka@pitt.edu
**Juan Manfredi**- Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
- MR Author ID: 205679
- Email: manfredi@pitt.edu
- Received by editor(s): January 20, 2016
- Received by editor(s) in revised form: April 17, 2016, and April 22, 2016
- Published electronically: May 11, 2017
- Additional Notes: The first and second authors were partially supported by NSF award DMS-1406730
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 7387-7403 - MSC (2010): Primary 35J92
- DOI: https://doi.org/10.1090/tran/6962
- MathSciNet review: 3683112